Experimental searches for supersymmetry (SUSY) are entering a new era. The failure to observe signals of sparticle production at the Large Hadron Collider (LHC) has eroded the central motivation for SUSY breaking at the weak scale. However, String Theory requires SUSY at the fundamental scale $M_s$ and hence SUSY could be broken at some high scale below $M_s$. Actually, if this were the case, the lack of experimental evidence for low-energy SUSY could have been anticipated, because most stringy models with high-scale SUSY breaking predict that sparticles would start popping up above about 10 TeV, well beyond the reach of current LHC experiments. We show that using next generation LHC experiments currently envisioned for the Forward Physics Facility (FPF) we could search for signals of neutrino-modulino oscillations to probe models with string scale in the grand unification region and SUSY breaking driven by sequestered gravity in gauge mediation. This is possible because of the unprecedented flux of neutrinos to be produced as secondary products in LHC collisions during the high-luminosity era and the capability of FPF experiments to detect and identify their flavors.